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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.inv_hyper.acosh"></a><a class="link" href="acosh.html" title="acosh">acosh</a>
</h3></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">acosh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">acosh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span>

<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">acosh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Computes the reciprocal of (the restriction to the range of <code class="literal">[0;+∞[</code>)
        <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the hyperbolic cosine
        function</a>, at x. Values returned are positive.
      </p>
<p>
        If x is in the range <code class="literal">]-∞;+1[</code> then returns the result of
        <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
      </p>
<p>
        The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
        type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and T otherwise.
      </p>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/acosh.svg" align="middle"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.inv_hyper.acosh.h0"></a>
        <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.accuracy"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.accuracy">Accuracy</a>
      </h5>
<p>
        Generally accuracy is to within 1 or 2 <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine
        epsilon</a> across all supported platforms.
      </p>
<h5>
<a name="math_toolkit.inv_hyper.acosh.h1"></a>
        <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.testing"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.testing">Testing</a>
      </h5>
<p>
        This function is tested using a combination of random test values designed
        to give full function coverage computed at high precision using the "naive"
        formula:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/acosh1.svg"></span>

        </p></blockquote></div>
<p>
        along with a selection of sanity check values computed using functions.wolfram.com
        to at least 50 decimal digits.
      </p>
<h5>
<a name="math_toolkit.inv_hyper.acosh.h2"></a>
        <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.implementation"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.implementation">Implementation</a>
      </h5>
<p>
        For sufficiently large x, we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/06/01/0001/" target="_top">approximation</a>:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/acosh2.svg"></span>

        </p></blockquote></div>
<p>
        For x sufficiently close to 1 we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/04/01/0001/" target="_top">approximation</a>:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/acosh4.svg"></span>

        </p></blockquote></div>
<p>
        Otherwise for x close to 1 we can use the following rearrangement of the
        primary definition to preserve accuracy:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/acosh3.svg"></span>

        </p></blockquote></div>
<p>
        Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/02/" target="_top">primary
        definition</a> is used:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/acosh1.svg"></span>

        </p></blockquote></div>
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      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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